# What is the discriminant of -x^2+10x-56=-4x-7?

Jul 27, 2015

For this quadratic, $\Delta = 0$.

#### Explanation:

In order to determine the determinant of this quadratic equation, you must first get it to quadratic form, which is

$a {x}^{2} + b x + c = 0$

For this general form, the determinant is equal to

$\Delta = {b}^{2} - 4 \cdot a \cdot c$

So, to get your equation to mthis form, add $4 x + 7$ to both sides of the equation

$- {x}^{2} + 10 x - 56 + \left(4 x + 7\right) = - \textcolor{red}{\cancel{\textcolor{b l a c k}{4 x}}} - \textcolor{red}{\cancel{\textcolor{b l a c k}{- 7}}} + \textcolor{red}{\cancel{\textcolor{b l a c k}{4 x}}} + \textcolor{red}{\cancel{\textcolor{b l a c k}{7}}}$

$- {x}^{2} + 14 x - 49 = 0$

Now identify what the values for $a$, $b$, and $c$ are. In your case,

$\left\{\begin{matrix}a = - 1 \\ b = 14 \\ c = - 49\end{matrix}\right.$

This means that the discriminant will be equal to

$\Delta = {14}^{2} - 4 \cdot \left(- 1\right) \cdot \left(- 49\right)$

$\Delta = 196 - 196 = \textcolor{g r e e n}{0}$

This means that your equation has only one real root

${x}_{1 , 2} = \frac{- b \pm \sqrt{\Delta}}{2 a}$

$x = \frac{- b \pm \sqrt{0}}{2 a} = \textcolor{b l u e}{- \frac{b}{2 a}}$

In your case, this solution is

$x = \frac{- 14}{2 \cdot \left(- 1\right)} = 7$